#471961

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,10,15,11,16,9,12)(7,17,18,29,21,28,20,13)(8,22,24,32,26,31,25,14)(19,34,23,30,35,27,33,36);(1,2,3,4)(6,12,16,15)(7,18,21,20)(14,22,32,31)(19,33,35,23);(2,4)(5,7)(6,13)(8,23)(9,18)(10,20)(11,21)(12,17)(14,30)(15,28)(16,29)(19,24)(22,34)(25,35)(26,33)(27,31)(32,36);(5,8,9,24,11,26,10,25)(6,14,15,31,16,32,12,22)(7,19,20,23,21,35,18,33)(13,27,17,30,29,34,28,36)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379157);